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Scheduling jobs with precedence constraints on a set of identical machines tominimize the total processing time (makespan) is a fundamental problem incombinatorial optimization. In practical settings such as cloud computing, jobsare often malleable, i.e., can be processed on multiple machinessimultaneously. The instantaneous processing rate of a job is a non-decreasingfunction of the number of machines assigned to it (we call it the processingfunction). Previous research has focused on practically relevant concaveprocessing functions, which obey the law of diminishing utility and generalizethe classical (non-malleable) problem. Our main result is a$(2+\epsilon)$-approximation algorithm for concave processing functions (forany $\epsilon > 0$), which is the best possible under complexity theoreticassumptions. The approximation ratio improves to $(1 + \epsilon)$ for theinteresting and practically relevant special case of power functions, i.e.,$p_j(z) = c_j \cdot z^{\gamma}$.

Precedence-constrained Scheduling of Malleable Jobs with Preemption